形式三角矩阵环的P_(1)-内射性和P_(1)-内射维数  

The P_(1)-injectivity and the P_(1)-injective Dimension of the Formal Triangular Matrix Rings

作　　者：谢晋 王芳贵 XIE Jin;WANG Fanggui(School of Mathematics and Physics,Mianyang Teachers’College,Mianyang 621000,Sichuan;School of Mathematical Sciences,Sichuan Normal University,Chengdu 610066,Sichuan)

机构地区：[1]绵阳师范学院数理学院,四川绵阳621000 [2]四川师范大学数学科学学院,四川成都610066

出　　处：《四川师范大学学报（自然科学版）》2021年第5期640-643,共4页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(11671283)。

摘　　要：设A、B为环,M为左B右A的双模,令T=(MA B0)为形式三角矩阵环.设R是任何环,右R-模D称为P_(1)-内射模,是指对任何投射维数不超过1的模P,有Ext1 R(P,D)=0.研究形式三角矩阵环T上P_(1)-内射模与P_(1)-内射维数.证明若M为平坦的左B-模,则T-模(Z,W)_(g)为P_(1)-内射模当且仅当Z_(A)与W_(B)为P_(1)-内射模;并证明若M为平坦的左B-模,则r.P_(1) dim(T)≤max{r.P_(1) dim(A),r.P_(1) dim(B)}.Let A,B be associative rings with identity,M be a left B-right A-bimodule and T be the formal triangular ring(A 0)M B.For any given ring R,an R-module D is called a P_(1)-injective module if Ext1 R(P,D)=0 for any R-module P with projective dimension at most 1.We study the P_(1)-injective modules and the P_(1)-injective dimension over the formal triangular matrix ring T.We prove that,if M is a left flat B-module,then a T-module(Z,W)_(g) is P_(1)-injective if and only if Z_(A),W_(B) are P_(1)-injective.We also prove that if r.P_(1) dim(T)≤max{r.P_(1) dim(A),r.P_(1) dim(B)}M is a left flat B-module.

关 键 词：P_(1)-内射模 P_(1)-内射维数 整体P_(1)-内射维数 形式三角矩阵环 

分 类 号：O154[理学—数学]